To analyze what happens to the x-intercepts of the function \( y = x^2 - 25 \) when the graph is reflected over the x-axis, we first need to understand what x-intercepts are. X-intercepts are points where the graph intersects the x-axis, which occurs when \( y = 0 \).
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Finding the x-intercepts of the original function:
- Set \( y = 0 \): \[ 0 = x^2 - 25 \]
- Solve for \( x \): \[ x^2 = 25 \] \[ x = 5 \quad \text{or} \quad x = -5 \] This means the x-intercepts are at \( (5, 0) \) and \( (-5, 0) \).
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Reflection over the x-axis:
- When we reflect the graph over the x-axis, the y-values are negated. Therefore, the new function after reflection would be \( y = -(x^2 - 25) = -x^2 + 25 \).
- Now, to find any new x-intercepts, we again set \( y = 0 \): \[ 0 = -x^2 + 25 \]
- Solve for \( x \): \[ x^2 = 25 \] \[ x = 5 \quad \text{or} \quad x = -5 \] The x-intercepts remain the same, at \( (5, 0) \) and \( (-5, 0) \).
Thus, when the graph of \( y = x^2 - 25 \) is reflected over the x-axis, the x-intercepts remain the same.
Correct response: The x-intercepts remain the same because the y-values get negated during a reflection over the x-axis.